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Given, A is a symmetric matrix and \[{\text{n}} \in {\text{N}}\],

As, A is a symmetric matrix, i.e. \[{\text{A = }}{{\text{A}}^{\text{T}}}\].

Now, take transpose of \[{{\text{A}}^{\text{n}}}\]

\[ \Rightarrow {{\text{(}}{{\text{A}}^{\text{n}}}{\text{)}}^{\text{T}}}\]

As we know that \[{{\text{(}}{{\text{A}}^{\text{n}}}{\text{)}}^{\text{T}}}{\text{ = (}}{{\text{A}}^{\text{T}}}{{\text{)}}^{\text{n}}}\]

\[ \Rightarrow {{\text{(}}{{\text{A}}^{\text{T}}}{\text{)}}^{\text{n}}}\]

As we know that \[{\text{A = }}{{\text{A}}^{\text{T}}}\]

So, \[{{\text{(}}{{\text{A}}^{\text{T}}}{\text{)}}^{\text{n}}} = {{\text{A}}^{\text{n}}}\]

Hence, \[{{\text{(}}{{\text{A}}^{\text{n}}}{\text{)}}^{\text{T}}}{\text{ = }}{{\text{A}}^{\text{n}}}\].

Hence, \[{{\text{A}}^{\text{n}}}\]is also a symmetric matrix.